Properties

Label 2-1350-1.1-c3-0-61
Degree $2$
Conductor $1350$
Sign $-1$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s − 14·7-s + 8·8-s + 3·11-s − 47·13-s − 28·14-s + 16·16-s + 39·17-s + 32·19-s + 6·22-s + 99·23-s − 94·26-s − 56·28-s + 51·29-s + 83·31-s + 32·32-s + 78·34-s − 314·37-s + 64·38-s − 108·41-s − 299·43-s + 12·44-s + 198·46-s − 531·47-s − 147·49-s − 188·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.0822·11-s − 1.00·13-s − 0.534·14-s + 1/4·16-s + 0.556·17-s + 0.386·19-s + 0.0581·22-s + 0.897·23-s − 0.709·26-s − 0.377·28-s + 0.326·29-s + 0.480·31-s + 0.176·32-s + 0.393·34-s − 1.39·37-s + 0.273·38-s − 0.411·41-s − 1.06·43-s + 0.0411·44-s + 0.634·46-s − 1.64·47-s − 3/7·49-s − 0.501·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 - 3 T + p^{3} T^{2} \)
13 \( 1 + 47 T + p^{3} T^{2} \)
17 \( 1 - 39 T + p^{3} T^{2} \)
19 \( 1 - 32 T + p^{3} T^{2} \)
23 \( 1 - 99 T + p^{3} T^{2} \)
29 \( 1 - 51 T + p^{3} T^{2} \)
31 \( 1 - 83 T + p^{3} T^{2} \)
37 \( 1 + 314 T + p^{3} T^{2} \)
41 \( 1 + 108 T + p^{3} T^{2} \)
43 \( 1 + 299 T + p^{3} T^{2} \)
47 \( 1 + 531 T + p^{3} T^{2} \)
53 \( 1 + 564 T + p^{3} T^{2} \)
59 \( 1 - 12 T + p^{3} T^{2} \)
61 \( 1 - 230 T + p^{3} T^{2} \)
67 \( 1 - 4 p T + p^{3} T^{2} \)
71 \( 1 - 120 T + p^{3} T^{2} \)
73 \( 1 + 1106 T + p^{3} T^{2} \)
79 \( 1 + 739 T + p^{3} T^{2} \)
83 \( 1 + 1086 T + p^{3} T^{2} \)
89 \( 1 + 120 T + p^{3} T^{2} \)
97 \( 1 - 1642 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.866399956819512231931710986603, −7.85629775974044779333710572127, −6.98591666961703626990597599316, −6.40818764460253507737191620262, −5.31002910472011315545124437443, −4.71260857600479750760909332844, −3.44202701952137217735425207239, −2.87369228441141289965083776799, −1.53129210161528466072454122651, 0, 1.53129210161528466072454122651, 2.87369228441141289965083776799, 3.44202701952137217735425207239, 4.71260857600479750760909332844, 5.31002910472011315545124437443, 6.40818764460253507737191620262, 6.98591666961703626990597599316, 7.85629775974044779333710572127, 8.866399956819512231931710986603

Graph of the $Z$-function along the critical line